The Problem Solving and Communication Activity Series, 2008-2009

This Math Forum program is designed to help students expand their repertoire of ways to approach challenging problems. Over the course of the year, we covered many of the fundamental problem solving strategies in conjunction with the Problems of the Week. Please visit the Activity Series main page for an overview of the program.

This page shows the alignments of PoWs to problem-solving strategies during the first year we offered the activity series. Each problem focuses on a particular strategy in developmental sequence, and we provide a document describing activities to do with your students and examples of typical student responses to the problem. The problem-specific documents both illustrate the activities and help you anticipate ideas that might come up in your class. We cycle through the strategies multiple times during the year so that students can get better at problem solving over the course of the Activity Series and the school year.

The documents below are all in PDF format. If you have a Class membership or higher, you can see all of the supporting documents for all of the 2008-09 PoWs, including links to the problems, at the 2008-2009 PoW Teacher Resources Page, also known as the Teacher Support pages.

Round 0: Introducing the Activity Series

The introductory document contains a series of activities designed to get students thinking about what good problem solvers do, and how to communicate their thinking by writing (or talking) mathematically.

Round 1: Understanding the Problem

What does it mean to fully understand a problem, and how does it help students find solution paths and build confidence? Included in these documents are several activities that support students to develop strategies for understanding challenging math problems, along with facilitation suggestions for teachers.

Round 3: Guess and Check

Guess and check is an important (and popular) problem-solving strategy, though it often gets a bad rap and may not
be developed into the strong and powerful resource it could be. The guess and check strategy has at least three
purposes: (1) to understand a problem thoroughly, (2) to home in on a solution, and (3) to discover efficient ways to
jump to a solution by noticing patterns and developing related algebraic representations.

Round 4: Solve a Simpler Problem

Solve a Simpler Problem is a technique that can be used in several ways to solve challenging problems. In some situations you can see how to work the problem with easier numbers. This may show you an approach that you can try with the more difficult numbers. Second, you can choose to break the original problem into smaller steps, finding answers for parts of the problem, and then putting those together for the whole solution. Finally, students may see a way to change this hard problem into one that they have solved before.

Round 5: Making a Table

The Tables and Patterns strategy is a way to organize your problem solving that makes it easier to explore patterns in the calculations and results. It is often used after some initial work on the problem using Understanding the Problem or Guess and Check strategies. Tables can be used to efficiently home in on answers, or you can use tables to organize the logic of your calculations and make explicit the relationships between quantities in the problem. As you may have seen in the Simpler Problem strategy, tables can help put different iterations in order and compare them. Tables can take the form of simple t-tables to very complex spreadsheets. Spreadsheets and other related software are especially efficient because they can be used to rearrange your work for different comparisons without having to write it all over again.

Round 6: Understanding the Problem (Revisited)

What does it mean to fully understand a problem, and how does it help students find solution paths and build confidence? This time around we have expanded activities to elicit relevant knowledge, recognize implications, and make drawings that focus on the key mathematical information.

Round 7: Guess and Check (Revisited)

In our previous round of Guess and Check, we focused on using guess and check to understand the problem and to home in on a solution. In this round we delve deeper into the uses of guess and check and present activities that help students:

Round 8: Solve a Simpler Problem (Revisited)

Solve a Simpler Problem is a technique that can be used in several ways to solve challenging problems. In some situations you can see how to work the problem with easier numbers. This may show you an approach that you can try with the more difficult numbers. Second, you can choose to break the original problem into smaller steps, finding answers for parts of the problem, and then putting those together for the whole solution. Finally, students may see a way to change this hard problem into one that they have solved before.

Round 9: Making a Table (Revisited)

The Tables and Patterns strategy is a way to organize your problem solving that makes it easier to explore patterns in the calculations and results. It is often used after some initial work on the problem using Understanding the Problem or Guess and Check strategies. Tables can be used to efficiently home in on answers, or you can use tables to organize the logic of your calculations and make explicit the relationships between quantities in the problem. As you may have seen in the Simpler Problem strategy, tables can help put different iterations in order and compare them. Tables can take the form of simple t-tables to very complex spreadsheets. Spreadsheets and other related software are especially efficient because they can be used to rearrange your work for different comparisons without having to write it all over again.

Round 10: Cases

Case-based reasoning helps problem solvers to understand the problem, work towards a solution, surface
interesting mathematics, and verify the robustness of their solutions. To understand the problem, problem solvers
might test interesting or representative cases and think about the different outcomes they see. When solving the
problem, they might use cases to consider when certain outcomes will occur, or to narrow down the possibilities
they have to investigate. Some problems have different answers for different cases. Exploring different cases can
lead to questions that problem solvers might explore further, like, "what would happen if I used a negative number?"
or, "would this work for obtuse triangles, too?" Finally, when determining whether a possible solution is correct,
good problem solvers test their solution using multiple cases, especially cases that they know behave differently.

Round 11: Logical Reasoning

Logic is an inherent part of the mathematical problem solving process and was used in some ways through our Activity Series already. However, some problems depend more on logic than on purely mathematical manipulations. Logic can help us find solutions when it looks as if we are unable to solve them based on our equations. Even with problems that are primarily solved through calculations, the questions and techniques of logical reasoning can help us organize and find efficient solution approaches to problems. In this sense Logical Reasoning is particularly useful in combination with approaches such as "Noticing and Wonderings" or "PoW IQ" from Understanding the Problem (Rounds 1, 2, or 6).

Round 12: Change the Representation

All math problems, whether they are word problems, arithmetic problems, equations to solve, etc., come to us in a particular representation. Word problems are represented in story form, using words and often referencing a particular context. Arithmetic problems are represented numerically. Equations are represented using mathematical symbols. Each representation has benefits to the problem solver. For example, word problems allow students to apply their knowledge of the given context, which can allow them to check that their approaches are reasonable.
Numeric and symbolic representations can make it easy for students to manipulate objects in the problem, and to quickly see patterns. Visual and physical representations, such as manipulatives, diagrams, and graphs, can often
help students gain new insights into the problem and provide them with additional tools for solving it. Changing the representation can mean use of a different form of representation (e.g. using a line drawing for a word problem) or it can mean trying different ways of presenting the information in the same form (e.g. rewriting all of the numbers as
fractions with a numerator of 1). Considering multiple representations and choosing representations that fit the
problem well are important problem-solving skills.

Round 13: Make a Mathematical Model

A mathematical model is a way to describe a situation, usually real-world, using numeric and mathematical relationships. Mathematical models usually have inputs, operations on those inputs, certain parameters or constants that make the operations fit the particular situation, and outputs that result from performing the operations on the inputs. Sometimes in problem solving, coming up with the mathematical model to use is at the heart of the problem. Problem-solvers are engaged in noticing quantities and relationships, selecting operations to describe the relationships, and fitting those operations to the specific scenario by setting parameters. Other times, the operations and relationships are given in the problem, and the problem solver's job is to organize the information and apply it to determine a final answer. In either case, identifying quantities and relationships, and recording information as mathematically as possible are key components of making mathematical models.

Round 14: Working Backwards

Working backwards is a particularly useful problem-solving strategy when you can clearly define the goal or end state of the problem, and you know a sequence of operations that were used in the problem. Reversing the operations and working backwards from the goal helps problem-solvers to describe the initial conditions or the most efficient path to the goal state. Working backwards is often applied to logic problems, like the famous one about crossing the river with a cabbage, a goat, and a wolf, in which you know the goal state (everything on the opposite side of the river), and you know what the legal moves are (rowing one animal across the river without leaving the cabbage with the goat or the wolf with the goat). You can work backwards from the goal, asking yourself, what must have been the last animal rowed across the river? What must have happened just before that?

Another sort of problem that working backwards can be applied to is a problem involving operations on a quantity, in which you know the final outcome after all the operations have been applied, and you need to find out the initial quantity. In these sorts of problems, it's useful to play the situation backwards, performing the inverse of each operation on the known, ending quantity until you are left with the initial quantity. This process of inverting operations is very similar to the process of solving an algebraic equation by "undoing" what's been done to the variable.

Round 15: Planning and Reflecting

As students get comfortable with more strategies, they begin to recognize and use multiple approaches in their problem-solving process. Following Polya (1945), it is common to break the problem-solving process down into four phases: Understanding the Problem, Making a Plan, Carrying Out the Plan, and Checking/Reflecting. Two of the most common issues in problem solving are (1) forgetting part of the problem or the ideas you have discovered that might be useful and (2) getting stuck, trying the same thing over and over without making new progress. Planning and reflecting can help you both solve problems and learn from your experience. What does it mean to get good at making a plan? How do I know when to start carrying out the plan? How do I know if I'm on a dead-end path? How do I effectively check my work?

Round 16: Getting Unstuck

If you never get stuck, then you are not solving interesting enough problems. Getting stuck (and, we hope, getting "unstuck"), is at the heart of problem solving. Challenging problems require that they be represented in many ways, approached with a variety of strategies, and checked again and again and again.

Round 17: Play

When students do the Noticing/Wondering activity, we often have them try to group their noticings into "quantities"
and "relationships". With a little practice, students get adept at finding the quantities and the relationships that are
explicitly stated in the problem. However, interesting math problems usually have deeper layers of relationships that
only emerge as problem solvers "play" with the relationships and quantities.

In the recent activities focusing on Planning and Getting Unstuck, we began to highlight some of the phases of
problem solving, and to show how many of the activities in this series can be used to explore relationships as you
begin problem solving or if you get stuck along the way.

Continuing in this vein, Round 17 focuses on some of the ways problem solvers play with relationships and explore
patterns before they delve deeply into a single problem-solving strategy. In order to make clear different aspects of
problem solving, we've broken the "play" process out somewhat artificially - expert problem solvers move back and
forth fluidly between understanding the problem, playing with relationships, and carrying out strategies. However,
for purposes of illustration, we think it will be useful to focus on those phases separately.

Round 19: Make a Mathematical Model (Revisited)

Round 20: Understanding the Problem - Wondering

In our third time through Understanding the Problem, we focus on wondering - on those questions problem solvers ask themselves as they try to understand the problem and find a solution path. It is usually the wondering that leads us to a solution. First we notice. Then we wonder about meaning and implications and possibilities. Then we pursue those questions and work our way to a solution.

As we've worked on noticing and wondering with students, we've noticed that their wondering is often interesting but could be more mathematical. Persistence is an important part of the process, for instance investigating aspects we wonder about, even if they don't seem promising. It helps to have a repertoire of reflective questions (wonderings) that help us get unstuck, connect to prior knowledge, and use our noticings to solve the problem. So, this round we have shared a lot of questions that have been useful to us.